Lossless Power Transfer

A very important equation in electric power system analysis is the so-called "power angle equation". It relates the power flow across a largely reactive transmission line to the RMS voltage magnitudes at the two ends, the voltage angles at the two ends, and the inductive reactance in between. Suppose the voltage at one end of the line is $V_1 \angle \theta_1$ , the voltage at the other end is $V_2 \angle \theta_2$ , and the inductive reactance is $X$ . The power flow is:

$P = \frac{V_1 V_2}{X} sin(\theta_1 - \theta_2)$

Plugging in numbers from this problem:

$P = \frac{3 \times 5}{4} sin(\pi/6 - 0) = \frac{15}{8} = \boxed{1.875}$

We can also simulate the circuit in the time-domain and integrate the instantaneous power over an integer number of cycles, and then divide by the same number of cycles. This yields the same result, but is somewhat more involved.